Square roots, and the prime-factor test for a perfect square

0:00Here is a square, and its area is forty-nine. Not its side. Its area. So what is the side? Seven, because seven sevens are forty-nine. That is all a square root is. You are running a squaring backwards. Squaring takes a side and hands you an area. A square root takes the area and hands you back the side. So a root is not a strange operation. It is a length.

0:30There is a catch, and it turns up straight away. Eight times eight is sixty-four. But minus eight times minus eight is also sixty-four, because a negative times a negative comes out positive. So sixty-four has two square roots. Eight, and minus eight. Every square number does, one of each sign. Nought is the only one with a single root. There is a sign for a square root, and from here on we will let it mean the positive one.

1:03That is a decision, not a discovery. The minus eight has not gone anywhere. We are agreeing not to write it. Now two numbers. Five hundred and seventy-six, and three hundred and twenty-seven. One of them is settled with no work at all. A square can only end in six of the ten digits. Nought, one, four, five, six, nine. Three hundred and twenty-seven ends in seven. So it is not a square, and we are finished with it.

1:37Five hundred and seventy-six ends in six, which is allowed. But allowed is not the same as yes. One hundred and fifty-six ends in six too, and that is not a square either. A last digit can throw a number out. It can never let one in. So how do we settle five hundred and seventy-six? The obvious way. Write squares out until you get there. Twenty squared is four hundred. Twenty-one squared, four hundred and forty-one. Twenty-two, four hundred and eighty-four.

2:14Twenty-three, five hundred and twenty-nine. Twenty-four. Five hundred and seventy-six. There it is, so the root is twenty-four. That worked. But notice the cost. We walked all the way up to twenty-four, and the bigger the number, the longer the walk. There is a second way, and it is prettier. Take eighty-one and subtract the odd numbers from it, in order. Eighty-one take one is eighty. Take three, seventy-seven. Take five, seventy-two.

2:49Seven, sixty-five. Nine, fifty-six. Eleven, forty-five. Thirteen, thirty-two. Fifteen, seventeen. Seventeen, nothing. It landed exactly on nought, so eighty-one is a square. And count the subtractions. Nine of them, and nine is the root. That is a lovely test. But it is the same bargain: seven hundred and twenty-nine takes twenty-seven steps. So we have two tests, and they have the same shape. Each one walks, and the walk is about as long as the root.

3:27For a three-digit number that is nothing. For a six-digit one you are counting for a very long time. So stop asking what the number is near. Ask what it is made of. Every whole number breaks into primes. Three hundred and twenty-four breaks into two, two, three, three, three, three. Six primes. Now deal them out like cards into two piles, one at a time, turn and turn about.

3:58Two, left. Two, right. Three, left. Three, right. Three, left. Three, right. Now look at the piles. Two, three, three. And two, three, three. They are identical. Multiply one of them out. Two threes are six, and six threes are eighteen. Eighteen. And eighteen eighteens are three hundred and twenty-four. The root came out of the pile. We never went looking for it. So why does that work? Not why did it work here. Why it has to.

4:30Suppose a number is something times itself. Then whatever primes that something is built from, the number has every one of them twice, once from each copy. So every prime turns up an even number of times, and that is exactly what lets you split them into two matching piles. It runs the other way too. If every prime turns up an even number of times, the piles have to come out the same, and one pile times the other gives you the number back.

5:03But something is holding all of that up, and it is easy to walk straight past. A number has only one set of primes. Only one. Break it apart however you like, in any order, and you land on the same primes every time. That is what makes this a test instead of a lucky arrangement. If a number had two different sets of primes, it could pass the pairing in one of them and fail in the other, and the answer would depend on how you did the breaking.

5:35It does not. So the piles decide it. Now one that fails. One hundred and fifty-six. Two, two, three, thirteen. Deal them out. Two and three on the left, two and thirteen on the right. The piles are different, so it is not a square. And look at which primes are stranded. Not only the thirteen. The three as well. Both of them turn up once. It is tempting to say it failed because thirteen is big. It did not.

6:08Thirteen twice over is one hundred and sixty-nine, and that is a perfect square. Twelve is two, two, three, and it fails on the three. Size has nothing to do with it. Odd tallies do. Here is what the walking tests cannot even ask. Nine thousand four hundred and eight. Not a square, and either walk would tell you so, eventually. But break it up. Six twos, one three, two sevens.

6:39The twos pair off. The sevens pair off. The three is on its own. So we know precisely what is missing. It wants another three. Multiply by three and you get twenty-eight thousand two hundred and twenty-four, which is a square, with a root of one hundred and sixty-eight. Multiply, mind. Not remove. Taking the three out lands on a square as well, but on a different number, and nobody asked for a different number.

7:11And when the answer really is no, you can still pin the number down. One thousand nine hundred and thirty-six. Forty squared is one thousand six hundred. Fifty squared is two thousand five hundred. So the root is somewhere between forty and fifty. It ends in six, and a square only ends in six when its root ends in four or in six. So it is forty-four, or forty-six. Try the one in between. Forty-five squared.

7:43Careful here. It is not one thousand six hundred plus twenty-five. Forty and five, squared, gives one thousand six hundred, plus twenty-five, plus two lots of forty fives. Four hundred more. Two thousand and twenty-five. That has overshot, so the root is below forty-five, and forty-six is out. Forty-four. And forty-four squared is one thousand nine hundred and thirty-six. When there is no root at all, the bracket is the answer.

8:17Two hundred and fifty. Fifteen squared is two hundred and twenty-five, sixteen squared is two hundred and fifty-six. So it lies between fifteen and sixteen, and much nearer sixteen. Six above it, twenty-five below. Here is what that is good for. Someone has a square of cloth with an area of one hundred and twenty-five, and wants to cut a fifteen by fifteen square out of it. One hundred and twenty-five sits between eleven squared, one hundred and twenty-one, and twelve squared, one hundred and forty-four.

8:56So the biggest whole square that fits has a side of eleven. Not fifteen. Fifteen would need two hundred and twenty-five. That question is now completely answered, and no exact root appeared anywhere in it. One last picture. A panel of motifs, nine across and nine down. Every motif is a block of tiny squares, five by five. So how many tiny squares are there altogether? Eighty-one motifs, twenty-five in each.

9:31Two thousand and twenty-five. Which we have already met. It is forty-five squared, the number that turned up while we were bracketing. And it had to be a square. Eighty-one is a square, twenty-five is a square, and a square times a square is always a square. But take the total apart and you never need to notice that. Three, three, three, three, five, five. Deal them out. Three, three, five. And three, three, five.

10:01Forty-five. Two matching piles, and the root falls out.